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Unformatted text preview: 8.5 Normal Distributions We have seen that the histogram for a binomial distribution with n = 20 trials and p = 0.50 was shaped like a bell if we join the tops of the rectangles with a smooth curve. Real world data, such as IQ scores, weights of individuals, heights, test scores have histograms that have a symmetric bell shape. We call such distributions Normal distributions. This will be the focus of this section. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/De_Moivre.html Three mathematicians contributed to the mathematical foundation for this curve. They are Abraham De Moivre, Pierre Laplace and Carl Frederick Gauss De Moivre pioneered the development of analytic geometry and the theory of probability . He published The Doctrine of Chance in 1718. The definition of statistical independence appears in this book together with many problems with dice and other games. He also investigated mortality statistics and the foundation of the theory of annuities DeMoivre Laplace Laplace also systematized and elaborated probability theory in &quot;Essai Philosophique sur les Probabilits&quot; (Philosophical Essay on Probability, 1814). He was the first to publish the value of the Gaussian integral , Bell shaped curves Many frequency distributions have a symmetric, bell shaped histogram. For example, the frequency distribution of heights of males is symmetric about a mean of 69.5 inches. Example 2: IQ scores are symmetrically distributed about a mean of 100 and a standard deviation of 15 or 16. The frequency distribution of IQ scores is bell shaped. Example 3: SAT test scores have a bell shaped , symmetric distribution. Graph of a generic normal distribution 0.1 0.2 0.3 0.4 0.5-4-2 2 4 Series1 Values on X axis represent the number of standard deviation units a particular data value is from the mean. Values on the y axis represent probabilities of the random variable x. 0.1 0.2 0.3 0.4 0.5-4-2 2 4 Series1 Area under the Normal Curve 1. Normal distribution : a smoothed out histogram 2. P( a &lt; x &lt; b) = Probability that the random variable x is between a and b is determined by the area under the normal curve between x = a and x = b . Properties of Normal distributions 1. Symmetric about its mean, 2. Approaches, but not touches, the horizontal axis as x gets very large ( or x gets very small) 3. Almost all observations lie within 3 standard deviations from the mean. Area under normal curve Example: A midwestern college has an enrollment of 3264 female students whose mean height is 64.4 inches and the standard deviation is 2.4 inches. By constructing a relative frequency distribution, with class boundaries of 56, 57, 58, 74, we find that the frequency distribution resembles a bell shaped symmetrical distribution. Heights of Females at a College ( Relative frequency distribution with class width = 1 is smoothed out to form a normal, bell-shaped curve) .. Normal curve areas...
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